Problem: Eric made two investments: Investment $\text{Q}$ has a value of $\$500$ at the end of the first year and increases by $\$45$ per year. Investment $\text{R}$ has a value of $\$400$ at the end of the first year and increases by $10\%$ per year. Eric checks the value of his investments once a year, at the end of the year. What is the first year in which Eric sees that investment $\text{R}$ 's value exceeded investment $\text{Q}$ 's value?
Solution: Notice that investment $\text{Q}$ 's value grows linearly while investment $\text{R}$ 's value grows exponentially. This means investment $\text{R}$ 's value is bound to exceed investment $\text{Q}$ 's value at some point. Let's start calculating the value of each investment to see when that happens. Year Investment $\text{Q}$ Investment $\text{R}$ (Add $\$45$ each year.) (Multiply by $1.1$ each year.) $1$ $500$ $400$ $2$ $545$ $440$ $3$ $590$ $484$ $4$ $635$ $532.4$ $5$ $680$ $585.64$ $6$ $725$ $644.20$ $7$ $770$ $708.62$ $8$ $815$ $779.49$ $9$ $860$ $857.44$ $10$ $905$ $943.18$ In conclusion, investment $\text{R}$ 's value will first exceed investment $\text{Q}$ 's value in year number $10$.